On Tue, 2 Sep 2003, Stan Brown wrote in part: SB> I started to write that at first, but I don't think it's true. SB> The scores on the test may very well be continuous.
DB> You were right the first time, Stan. It is true: scores on a test DB> are NOT continuous. Ever. > Sorry, I don't buy that. Think "partial credit". Scores on an esay > exam or an exam that involves working problems can be subdivided and > are not necessarily integers. They can be subdivided indefinitely -- > in principle. And in principle this subdivision only gets you to the set of rational numbers, not the set of real numbers; and the rational numbers are isomorphic to the set of integers. > (In practice I feel pretty confident that no grading > protocol would go to the ten-thousandth of a point.) THAT's a safe bet, I bet! DB> (1) Scores on a test are nearly always the number of items correct, DB> or a (usually linear) transformation thereof; and the number of DB> items right is necessarily an integer, not a continuous measure. > > Sure, for a multiple-choice test. I apologize if I missed it, but I > think we were told just "test", not multiple-choice test. Doesn't matter. I never use multiple-choice tests, and tend to mark answers to individual questions out of 10 for convenience of reporting partial scores to students. But the maximum score is still an integer (maybe 170 or 240 or whatever, but still). DB> (2) Even if the scores are transformed (via norms tables or DB> otherwise) to some kind of "scaled score", there are a finite number DB> of scores, not exceeding the number of items on the test nor the DB> number of examinees in the data set. > True, but I don't see the relevance. OK. Arguable, but not worth arguing... > If you have a sample of 10 deer mice and record their weights, do > you consider those weights _not_ part of a continuous distribution? Depends on what you mean by "those weights". I may well wish to consider the underlying concept of mass (quibble: not "weight", which refers to a force) as continuous. But the actual values one has as empirical measures are not: they are measured and reported to the nearest unit of convenient size for the purpose. (Humans are ordinarily measured to the nearest pound, in the U.S.; I'm not sure what the practice is in countries using the metric system, a kilogram being more than twice as much as a pound. Deer mice would surely be weighed at least to the nearest ounce, or perhaps to the nearest gram, and probably not to the nearest milligram. The units are ordinarily chosen so as to yield 2- or 3-digit precision in the range of values of interest.) DB> (3) The scores are stored and and manipulated using digital DB> computers; it is not possible for a digital instrument to deal with DB> a continuous quantity except by approximation, and ALWAYS to a DB> finite number of significant digits of precision. This would be DB> true EVEN IF the original measuring instrument reported values on a DB> continuum. > > By that argument, there is no such thing as a continuous > distribution or continuous data. There are such things as continuous distributions, but they are mathematical constructs. There is no such thing as continuous data, since data rely on (necessarily imperfect) measurement. > Sure, the universe is inherently digital and discrete (as we known > from the quantum theory), but for all practical purposes ordinary > things are continuous. Would you seriously consider weight or time > to be discrete random variables just because the computer has only > 18 digits of accuracy? Surely not! See above. I think it fair to distinguish between a random variable (which, viewed as a latent variable, may be continuous) and a realization of a random variable (which, existing in the physical world, and measured to finite precision, is not continuous). > These points look so obvious that I fear either I'm completely > misunderstanding you or we're working from incompatible definitions. Sounds as though you're understanding me well enough, but it is possible that we're using not altogether compatible definitions... Ciao! -- Don. ----------------------------------------------------------------------- Donald F. Burrill [EMAIL PROTECTED] 56 Sebbins Pond Drive, Bedford, NH 03110 (603) 626-0816 . . ================================================================= Instructions for joining and leaving this list, remarks about the problem of INAPPROPRIATE MESSAGES, and archives are available at: . http://jse.stat.ncsu.edu/ . =================================================================
